Quadratic derivative nonlinear Schrödinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&;t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&;x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$ where the quadratic nonlinearity has the form ${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }We study the global in time existence of small classical solutions to the nonlinear Schr?dinger equation with quadratic interactions of derivative type in two space dimensions

$\left\{{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \right.\quad\quad\quad\quad\quad\quad (0.1)$\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)      2. A remark on global existence of solutions of shadow systems      Tuoc Van Phan 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,4(1):395-400 Let Ω be an open, bounded domain in \mathbbRn  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d 1, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system: $ \left \{ {l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \right. $ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right.      3. A note on a fourth order degenerate parabolic equation in higher space dimensions      Junjie Li 《Mathematische Annalen》2007,339(2):251-285 We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions       4. Existence Theory for an Arbitrary Order Fractional Differential Equation with Deviating Argument      Youhui Su  Zhaosheng Feng 《Acta Appl Math》2012,118(1):81-105 In this paper, we are concerned with the existence criteria for positive solutions of the following nonlinear arbitrary order fractional differential equations with deviating argument $\left \{{l@{\quad}l}D_{0^+}^{\alpha}u(t)+h(t)f(u(\theta(t)))=0, & t\in ( 0,1 ),\ n-1<\alpha\leq n,\\[3pt]u^{(i)}(0)=0, & i=0,1,2,\ldots,n-2,\\[3pt][D_{0^+}^{\beta} u(t)]_{t=1}=0, & 1\leq\beta\leq n-2, \right .$\left \{\begin{array}{l@{\quad}l}D_{0^+}^{\alpha}u(t)+h(t)f(u(\theta(t)))=0, & t\in ( 0,1 ),\ n-1<\alpha\leq n,\\[3pt]u^{(i)}(0)=0, & i=0,1,2,\ldots,n-2,\\[3pt][D_{0^+}^{\beta} u(t)]_{t=1}=0, & 1\leq\beta\leq n-2,\end{array} \right .      5. Linear Approximation and Asymptotic Expansion of Solutions in Many Small Parameters for a Nonlinear Kirchhoff Wave Equation with Mixed Nonhomogeneous Conditions      Le Thi Phuong Ngoc  Nguyen Thanh Long 《Acta Appl Math》2010,112(2):137-169 In this paper, we consider the following nonlinear Kirchhoff wave equation $\left\{\begin{array}{l}u_{tt}-\frac{\partial }{\partial x}(\mu (u,\Vert u_{x}\Vert ^{2})u_{x})=f(x,t,u,u_{x},u_{t}),\quad 0 (1) where \(\widetilde{u}_{0}\), \(\widetilde{u}_{1}\), μ, f, g are given functions and \(\Vert u_{x}\Vert ^{2}=\int_{0}^{1}u_{x}^{2}(x,t)dx.\) To the problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In particular, motivated by the asymptotic expansion of a weak solution in only one, two or three small parameters in the researches before now, an asymptotic expansion of a weak solution in many small parameters appeared on both sides of (1)1 is studied.       6. Fujita exponents for evolution problems with nonlocal diffusion   总被引：1，自引：0，他引：1   Jorge García-Melián  Fernando Quirós 《Journal of Evolution Equations》2010,10(1):147-161 We prove the existence of a critical exponent of Fujita type for the nonlocal diffusion problem $\left\{{l@{\quad}l}u_t(x, t) = J*u(x, t)-u(x, t) + u^p(x, t), & \qquad x \in \mathbb{R}^N,\; t > 0,\\ u(x, 0) = u_0(x), & \qquad x \in\mathbb{R}^N,\right.$\left\{\begin{array}{l@{\quad}l}u_t(x, t) = J*u(x, t)-u(x, t) + u^p(x, t), & \qquad x \in \mathbb{R}^N,\; t > 0,\\ u(x, 0) = u_0(x), & \qquad x \in\mathbb{R}^N,\end{array}\right.      7. A remark on global existence of solutions of shadow systems      Tuoc Van Phan 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,63(2):395-400 Let ?? be an open, bounded domain in ${\mathbb{R}^n\;(n \in \mathbb{N})}$ with smooth boundary ???. Let p, q, r, d 1, ?? be positive real numbers and s be a non-negative number which satisfies ${0 < \frac{p-1}{r} < \frac{q}{s+1}}$ . We consider the shadow system of the well-known Gierer?CMeinhardt system: $$ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right. $$ We prove that solutions of this system exist globally in time under some conditions on the coefficients. Our results are based on a priori estimates of the solutions and improve the global existence results of Li and Ni in [4].      8. On finite-time blow-up for a nonlocal parabolic problem arising from shear bands in metals      Gao-Feng Zheng 《Proceedings of the American Mathematical Society》2007,135(5):1487-1494 Results on finite-time blow-up of solutions to the nonlocal parabolic problem are established. They extend some known results to higher dimensions.       9. Continuous dependence on the boundary conditions for the Wentzell Laplacian      Giuseppe Maria Coclite  Angelo Favini  Gisèle Ruiz Goldstein  Jerome A. Goldstein  Silvia Romanelli 《Semigroup Forum》2008,77(1):101-108 The solution u of the well-posed problem depends continuously on (a ij ,β,γ,q). Dedicated to Karl H. Hofmann on his 75th birthday.      10. Uniqueness of Positive Solutions for a Nonlinear Elliptic System      Norihisa Ikoma 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(5):555-567 In this paper we study the uniqueness of nontrivial positive solutions for the following second order nonlinear elliptic system: $\left\{\begin{aligned} -\Delta u_1+V_1(|x|) u_1 &= \mu_1 u_1^3+\beta u_1u_2^2 & &\quad{\rm in} \ {\mathbb R}^{N},\\ -\Delta u_2+V_2(|x|)u_2&=\beta u_1^2u_2+\mu_2 u_2^3&&\quad{\rm in} \ {\mathbb R}^{N}.\end{aligned}\right.$\left\{\begin{aligned} -\Delta u_1+V_1(|x|) u_1 &= \mu_1 u_1^3+\beta u_1u_2^2 & &\quad{\rm in} \ {\mathbb R}^{N},\\ -\Delta u_2+V_2(|x|)u_2&=\beta u_1^2u_2+\mu_2 u_2^3&&\quad{\rm in} \ {\mathbb R}^{N}.\end{aligned}\right.      11. QUALITATIVE METHOD IN ANALYSIS OF A P-L-L WITH TANGENT CHARACTERISTIC AND FREQUENCY MODULATION INPUT      Wang Lian  Wang Muqiu 《数学年刊B辑(英文版)》1980,1(34):411-419 In analysis of p-L-L with tangent characteristic and frequency modulation input, we have obtained the following two types of the phase looked loop equation. \[\begin{array}{l} \frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + \alpha \frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t + {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (I)\\frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + (\alpha + \eta {\sec ^2}\varphi )\frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t - {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (II) \(\alpha > 0,\gamma > 0,\eta > 0,{\beta _1} > 0,{\beta _2} > 0,{\Omega _M} > 0) \end{array}\] In this paper, our aim is to explain the usual qualitative method and Lyapunov's function method, by which the existence of a periodic solution of (I), (II) is established. In addition, we especially point out: How is to construct the Lyapunovas function for the nonlinear and nonairtoiiomous system? This is a very important problem.      12. Remarks on the regularity criteria of the solutions of the 3D micropolar fluid equations          下载免费PDF全文 Liu Qiao 《Journal of Applied Analysis & Computation》2014,4(4):355-365 We provide two regularity criteria for the weak solutions of the 3D micropolar fluid equations, the first one in terms of one directional derivative of the velocity, i.e., $\partial_{3}u$, while the second one is is in terms of the behavior of the direction of the velocity $\frac{u}{|u|}$. More precisely, we prove that if \begin{equation*} \partial_{3}u \in L^{\beta}(0,T;L^{\alpha}(\mathbb{R}^{3}))\quad\text{ with }\frac{2}{\beta}+\frac{3}{\alpha}\leq 1+\frac{1}{\alpha}, 2&lt; \alpha \leq\infty, 2\leq\beta&lt; \infty; \end{equation*} or \begin{equation*} \operatorname{div}\left(\frac{u}{|u|}\right)\in L^{\frac{4}{1-2r}}(0,T;\dot{X}_{r}(\mathbb{R}^{3}))\quad \text{ with } 0\leq r&lt; \frac{1}{2}, \end{equation*} then the weak solution $(u(x,t),\omega(x,t))$ is regular on $\mathbb{R}^{3}\times [0,T]$. Here $\dot{X}_{r}(\mathbb{R}^{3})$ is the multiplier space.      13. Existence and Comparison Results for Non-Uniformly Parabolic Problems      Francesco Della Pietra  Giuseppina di Blasio 《Mediterranean Journal of Mathematics》2010,7(3):323-340 In this paper we prove existence and comparison results for nonlinear parabolic equations which are modeled on the problem $\left\{{ll}{u_t - {\rm div}\,\left(\frac{1}{(1+|u|)^{\alpha}}|Du|^{p-2}Du\right) =f\quad\hskip 2pt \,\,{\rm in}\,\Omega\times(0,T),}\\ {u=0\qquad\qquad\qquad\qquad\quad\quad\qquad{\rm on}\,\partial\Omega\times(0,T),}\\ {u(x,0)=u_0(x)\quad\qquad\qquad\qquad\qquad{\rm in}\,\Omega,}\right.$\left\{\begin{array}{ll}{u_t - {\rm div}\,\left(\frac{1}{(1+|u|)^{\alpha}}|Du|^{p-2}Du\right) =f\quad\hskip 2pt \,\,{\rm in}\,\Omega\times(0,T),}\\ {u=0\qquad\qquad\qquad\qquad\quad\quad\qquad{\rm on}\,\partial\Omega\times(0,T),}\\ {u(x,0)=u_0(x)\quad\qquad\qquad\qquad\qquad{\rm in}\,\Omega,}\end{array}\right.      14. Existence of at least three positive solutions for multi-point boundary value problem on infinite intervals with p-Laplacian operator      Xiangkui Zhao  Weigao Ge 《Journal of Applied Mathematics and Computing》2008,28(1-2):391-403 In this paper, we consider the multi-point boundary value problem of second-order nonlinear differential equation on a half line, $$\left\{\begin{array}{l@{\quad }l}(\phi_{p}(u'))'(t)+q(t)f(t,u(t),u'(t))=0,&0<t<\infty,\\[6pt]u'(0)=\sum_{i=1}^{m-2}\alpha_{i}u(\xi_{i}),&u'(\infty)=0.\end{array}\right.$$ By using a fixed point theorem due to Avery and Peterson, we show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term.      15. Forward dynamical problem for the Timoshenko beam      M. I. Belishev  A. L. Pestov 《Journal of Mathematical Sciences》2010,167(5):603-621 The initial boundary value problem $ {*{20}{c}} {\rho {u_{tt}} - {{\left( {\Gamma {u_x}} \right)}_x} + A{u_x} + Bu = 0,} \hfill & {x > 0,\quad 0 < t < T,} \hfill \\ {u\left| {_{t = 0}} \right. = {u_t}\left| {_{t = 0}} \right. = 0,} \hfill & {x \geq 0,} \hfill \\ {u\left| {_{x = 0}} \right. = f,} \hfill & {0 \leq t \leq T,} \hfill \\ $ \begin{array}{*{20}{c}} {\rho {u_{tt}} - {{\left( {\Gamma {u_x}} \right)}_x} + A{u_x} + Bu = 0,} \hfill & {x > 0,\quad 0 < t < T,} \hfill \\ {u\left| {_{t = 0}} \right. = {u_t}\left| {_{t = 0}} \right. = 0,} \hfill & {x \geq 0,} \hfill \\ {u\left| {_{x = 0}} \right. = f,} \hfill & {0 \leq t \leq T,} \hfill \\ \end{array}      16. The L p -Solvability of the Dirichlet Problem for Planar Elliptic Equations, Sharp Results      Carlo Sbordone  Gabriella Zecca 《Journal of Fourier Analysis and Applications》2009,15(6):871-903 Assume that the elliptic operator L=div (A(x)∇) is L p -resolutive, p>1, on the unit disc \mathbbD ì \mathbb R2\mathbb{D}\subset \mathbb {R}^{2} . This means that the Dirichlet problem $\left\{{l@{\quad}l}Lu=0&\mbox{in }\mathbb{D},\\[3pt]u=g&\mbox{on }\partial\mathbb{D}\right.$\left\{\begin{array}{l@{\quad}l}Lu=0&\mbox{in }\mathbb{D},\\[3pt]u=g&\mbox{on }\partial\mathbb{D}\end{array}\right.      17. The Cauchy–Kowalewski Theorem in the Space of Pseudo Q-holomorphic Functions      Sezayi Hızlıyel  Yeşim Sağlam Özkan 《Complex Analysis and Operator Theory》2016,10(5):953-963 In this work, we prove the Cauchy–Kowalewski theorem for the initial-value problem $$\begin{aligned} \frac{\partial w}{\partial t}= & {} Lw \\ w(0,z)= & {} w_{0}(z) \end{aligned}$$ where $$\begin{aligned} Lw:= & {} E_{0}(t,z)\frac{\partial }{\partial \overline{\phi }}\left( \frac{ d_{E}w}{dz}\right) +F_{0}(t,z)\overline{\left( \frac{\partial }{\partial \overline{\phi }}\left( \frac{d_{E}w}{dz}\right) \right) }+C_{0}(t,z)\frac{ d_{E}w}{dz} \\&+G_{0}(t,z)\overline{\left( \frac{d_{E}w}{dz}\right) } +A_{0}(t,z)w+B_{0}(t,z)\overline{w}+D_{0}(t,z) \end{aligned}$$ in the space \(P_{D}\left( E\right) \) of Pseudo Q-holomorphic functions.       18. Error function inequalities      Horst Alzer 《Advances in Computational Mathematics》2010,33(3):349-379 We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ \delta_{\alpha} < \frac{ \mbox{erf} \left( x+ \mbox{erf}(y)^{\alpha}\right) +\mbox{erf}\left( y+ \mbox{erf}(x)^{\alpha}\right) } {\mbox{erf}\left( \mbox{erf}(x)+\mbox{erf}(y)\right) } < \Delta_{\alpha} $$ with the best possible bounds $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$       19. The linear heat equation with highly oscillating potential      Ismail Kombe 《Proceedings of the American Mathematical Society》2004,132(9):2683-2691 In this paper we consider the following initial value problem: where and . Nonexistence of positive solutions is analyzed.       20. Positive Solutions for Second Order Singular Boundary Value Problems with Derivative Dependence on Infinite Intervals      Baoqiang Yan  Donal O’Regan  Ravi P. Agarwal 《Acta Appl Math》2008,103(1):19-57 The existence of at least one positive solution and the existence of multiple positive solutions are established for the singular second-order boundary value problem using the fixed point index, where f may be singular at x=0 and px′=0. The project is supported by the fund of National Natural Science (10571111) and the fund of Natural Science of Shandong Province.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

A remark on global existence of solutions of shadow systems

Let Ω be an open, bounded domain in \mathbbRⁿ  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d ₁, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system:

A note on a fourth order degenerate parabolic equation in higher space dimensions

We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions      

Existence Theory for an Arbitrary Order Fractional Differential Equation with Deviating Argument

In this paper, we are concerned with the existence criteria for positive solutions of the following nonlinear arbitrary order fractional differential equations with deviating argument

Linear Approximation and Asymptotic Expansion of Solutions in Many Small Parameters for a Nonlinear Kirchhoff Wave Equation with Mixed Nonhomogeneous Conditions

In this paper, we consider the following nonlinear Kirchhoff wave equation

$\left\{\begin{array}{l}u_{tt}-\frac{\partial }{\partial x}(\mu (u,\Vert u_{x}\Vert ^{2})u_{x})=f(x,t,u,u_{x},u_{t}),\quad 0

(1)

where \(\widetilde{u}_{0}\), \(\widetilde{u}_{1}\), μ, f, g are given functions and \(\Vert u_{x}\Vert ^{2}=\int_{0}^{1}u_{x}^{2}(x,t)dx.\) To the problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In particular, motivated by the asymptotic expansion of a weak solution in only one, two or three small parameters in the researches before now, an asymptotic expansion of a weak solution in many small parameters appeared on both sides of (1)₁ is studied.

     

Fujita exponents for evolution problems with nonlocal diffusion

We prove the existence of a critical exponent of Fujita type for the nonlocal diffusion problem

A remark on global existence of solutions of shadow systems

Let ?? be an open, bounded domain in ${\mathbb{R}^n\;(n \in \mathbb{N})}$ with smooth boundary ???. Let p, q, r, d ₁, ?? be positive real numbers and s be a non-negative number which satisfies ${0 < \frac{p-1}{r} < \frac{q}{s+1}}$ . We consider the shadow system of the well-known Gierer?CMeinhardt system: $$ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right. $$ We prove that solutions of this system exist globally in time under some conditions on the coefficients. Our results are based on a priori estimates of the solutions and improve the global existence results of Li and Ni in [4].     

On finite-time blow-up for a nonlocal parabolic problem arising from shear bands in metals

Results on finite-time blow-up of solutions to the nonlocal parabolic problem

are established. They extend some known results to higher dimensions.

     

Continuous dependence on the boundary conditions for the Wentzell Laplacian

The solution u of the well-posed problem

Uniqueness of Positive Solutions for a Nonlinear Elliptic System

In this paper we study the uniqueness of nontrivial positive solutions for the following second order nonlinear elliptic system:

QUALITATIVE METHOD IN ANALYSIS OF A P-L-L WITH TANGENT CHARACTERISTIC AND FREQUENCY MODULATION INPUT

In analysis of p-L-L with tangent characteristic and frequency modulation input, we have obtained the following two types of the phase looked loop equation. \[\begin{array}{l} \frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + \alpha \frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t + {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (I)\\frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + (\alpha + \eta {\sec ^2}\varphi )\frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t - {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (II) \(\alpha > 0,\gamma > 0,\eta > 0,{\beta _1} > 0,{\beta _2} > 0,{\Omega _M} > 0) \end{array}\] In this paper, our aim is to explain the usual qualitative method and Lyapunov's function method, by which the existence of a periodic solution of (I), (II) is established. In addition, we especially point out: How is to construct the Lyapunovas function for the nonlinear and nonairtoiiomous system? This is a very important problem.     

Remarks on the regularity criteria of the solutions of the 3D micropolar fluid equations

We provide two regularity criteria for the weak solutions of the 3D micropolar fluid equations, the first one in terms of one directional derivative of the velocity, i.e., $\partial_{3}u$, while the second one is is in terms of the behavior of the direction of the velocity $\frac{u}{|u|}$. More precisely, we prove that if \begin{equation*} \partial_{3}u \in L^{\beta}(0,T;L^{\alpha}(\mathbb{R}^{3}))\quad\text{ with }\frac{2}{\beta}+\frac{3}{\alpha}\leq 1+\frac{1}{\alpha}, 2&lt; \alpha \leq\infty, 2\leq\beta&lt; \infty; \end{equation*} or \begin{equation*} \operatorname{div}\left(\frac{u}{|u|}\right)\in L^{\frac{4}{1-2r}}(0,T;\dot{X}_{r}(\mathbb{R}^{3}))\quad \text{ with } 0\leq r&lt; \frac{1}{2}, \end{equation*} then the weak solution $(u(x,t),\omega(x,t))$ is regular on $\mathbb{R}^{3}\times [0,T]$. Here $\dot{X}_{r}(\mathbb{R}^{3})$ is the multiplier space.     

Existence and Comparison Results for Non-Uniformly Parabolic Problems

In this paper we prove existence and comparison results for nonlinear parabolic equations which are modeled on the problem

Existence of at least three positive solutions for multi-point boundary value problem on infinite intervals with p-Laplacian operator

In this paper, we consider the multi-point boundary value problem of second-order nonlinear differential equation on a half line, $$\left\{\begin{array}{l@{\quad }l}(\phi_{p}(u'))'(t)+q(t)f(t,u(t),u'(t))=0,&0<t<\infty,\\[6pt]u'(0)=\sum_{i=1}^{m-2}\alpha_{i}u(\xi_{i}),&u'(\infty)=0.\end{array}\right.$$ By using a fixed point theorem due to Avery and Peterson, we show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term.     

Forward dynamical problem for the Timoshenko beam

The initial boundary value problem

The L p -Solvability of the Dirichlet Problem for Planar Elliptic Equations, Sharp Results

Assume that the elliptic operator L=div (A(x)∇) is L ^p -resolutive, p>1, on the unit disc \mathbbD ì \mathbb R²\mathbb{D}\subset \mathbb {R}^{2} . This means that the Dirichlet problem

The Cauchy–Kowalewski Theorem in the Space of Pseudo Q-holomorphic Functions

In this work, we prove the Cauchy–Kowalewski theorem for the initial-value problem

$$\begin{aligned} \frac{\partial w}{\partial t}= & {} Lw \\ w(0,z)= & {} w_{0}(z) \end{aligned}$$

where

$$\begin{aligned} Lw:= & {} E_{0}(t,z)\frac{\partial }{\partial \overline{\phi }}\left( \frac{ d_{E}w}{dz}\right) +F_{0}(t,z)\overline{\left( \frac{\partial }{\partial \overline{\phi }}\left( \frac{d_{E}w}{dz}\right) \right) }+C_{0}(t,z)\frac{ d_{E}w}{dz} \\&+G_{0}(t,z)\overline{\left( \frac{d_{E}w}{dz}\right) } +A_{0}(t,z)w+B_{0}(t,z)\overline{w}+D_{0}(t,z) \end{aligned}$$

in the space \(P_{D}\left( E\right) \) of Pseudo Q-holomorphic functions.

     

Error function inequalities

We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ \delta_{\alpha} < \frac{ \mbox{erf} \left( x+ \mbox{erf}(y)^{\alpha}\right) +\mbox{erf}\left( y+ \mbox{erf}(x)^{\alpha}\right) } {\mbox{erf}\left( \mbox{erf}(x)+\mbox{erf}(y)\right) } < \Delta_{\alpha} $$ with the best possible bounds $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$      

The linear heat equation with highly oscillating potential

In this paper we consider the following initial value problem:

where and . Nonexistence of positive solutions is analyzed.

     

Positive Solutions for Second Order Singular Boundary Value Problems with Derivative Dependence on Infinite Intervals

The existence of at least one positive solution and the existence of multiple positive solutions are established for the singular second-order boundary value problem